Portfolio Variance from an Asset Covariance Matrix

Summary

The document addresses how average variance and average covariance relate to the variance of an equally weighted portfolio. Its central formula expresses portfolio variance using the full asset variance-covariance matrix and a vector of portfolio weights. With equal weights, the same calculation can be interpreted as combining individual asset variances with cross-asset covariances, which motivates the common decomposition into average variance and average covariance.

The response emphasizes the general matrix method rather than giving a worked calculation for five assets. To apply it, the investor needs the asset weights and the complete covariance matrix; pairwise correlations and standard deviations can be used to derive covariances. The reply notes that the word “average” can be misleading, since portfolio risk is fundamentally a weighted combination of variances and covariances rather than a single unqualified average. It provides no numerical example or discussion of estimation error, so users must construct and interpret the inputs for their own portfolio.

Key ideas

• Portfolio variance is calculated as the weight vector multiplied through the asset covariance matrix.
• Equal portfolio weights allow variance and cross-asset covariance contributions to be grouped into averages.
• Pairwise correlations and asset standard deviations provide the ingredients for covariance estimates.
• The meaning of average covariance depends on the weights and the set of asset pairs included.

Tags

# how to calculate avarage variance and avarage covariance


# how to calculate avarage variance and avarage covariance












I would like to figure out how to calculate `av.variance` and `av.cov`.

I know how to calculate portfolio variance( for large portfolio),

```
var(p)=1/n avarage variance+(1-1/n)avarage covariance,
```

If we have a pair of correlation of five asset and standard deviation of each five asset, how are we going to calculate `avarage covariance` and `avarage variance` so that we can put the data in above formula?

It confuse me because some questions `avarage covariance` and `avarage variance` are given.

thanks,i got you

## Answer by pincopallino (score 0, accepted)

https://quant.stackexchange.com/a/12722

$w$ is a vector of portfolio weights.

Portfolio variance $Var(r_p)$ is:

$$Var(r_p) = w^T V w$$ where $V$ is your asset variance-covariance matrix.

Let me know if this helps: in theory $w = 1/n$ where $n$ is the number of assets should lead to some sort of "average" covariance even if I think "averages" do not really apply to this context.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.